I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by
$$ \,dS = \sin^{n-1}(\theta_1) \sin^{n-2}(\theta_2)\cdots \sin(\theta_{n-1})\,d\theta_{1}\,d\theta_{2}\cdots \,d\theta_{n-1}\,d\theta_{n},$$ with $\theta_{n} \in [0,2\pi)$ and the other angles in $[0, \pi]$.
My question is: what should be the integration limits in
$$\text{Area} = \idotsint \limits_{?} \,dS$$ to computed this intersection surface?
Thanks in advance!